The present invention relates to a method of detecting chaotic structures in a given medium.
A chaotic structure is a three-dimensional region of a medium inside which the variations in intensity between neighbouring pixels of an image are especially anarchic. Such regions, regarded as being chaotic, may appear in any three-dimensional imaging, such as for example in medical imaging, seismic imaging; in the case of seismic imaging, they pertain to channel complexes.
When a seismic block is visualized in a given direction, the channel complex or zone exhibits particularly disorganized movements. In order to detect or highlight the channel complex, it has been proposed that the error in the optical flux at every point be calculated according to a model in which the velocity vector field is assumed to be constant locally. When the abovementioned method of detecting the channel complexes is implemented, it is found that the points situated outside the channel complex correspond well to the model used, since locally they all undergo almost the same movement. In this case, the error in the optical flux, which represents the deviation between the velocity obtained and the ideal velocity which is a velocity corresponding to a zero error, is small. Conversely, the points situated inside the channel complex have very different movements locally and the assumption of a uniform field then becomes inadequate. Accordingly, the error in the optical flux, calculated in such a complex, is considerably bigger. Such a behaviour or such a difference between the calculated flux errors makes it possible very simply to distinguish the channel complexes in the form of three-dimensional regions. However, the optical flux calculation and the resulting monodirectional error remain dependent on the direction of visualization which is chosen arbitrarily. To obtain more robust detection of channel complexes, it would be possible to envisage calculating the flux error in a large number of directions. However, it would not be conceivable to average the calculated flux errors by displaying a seismic survey in all directions on account of prohibitive calculation time.
The manner of calculating the monodirectional flux error is given below.
Consider a block of seismic images whose axes are x, y, z with a direction of visualization which coincides with the z axis, the said direction of visualization being defined by the transposed vector xcex94=[001]Tand the component of the local displacement U or optical flux in the direction xcex94 is unity. The model with constant velocity vector field estimates the transposed vector U=[u,v,1]Tby minimizing the sum W below by the method of least squares:                     W        =                              ∑                          i              ∈              F                                ⁢                      xe2x80x83                    ⁢                                    (                                                                    Ex                    i                                    ⁢                  u                                +                                                      Ey                    i                                    ⁢                  v                                +                                  Ez                  i                                            )                        2                                              (1)            
in which:
is a three-dimensional window with axes x, y, z, of dimensions N*N*N, N being the number of pixels of the image along each of the three axes x, y, z.
Exi, Eyi, Ezi are the partial derivatives of the luminous intensity at the point i along the axes x, y and z.
The components u and v are calculated by the following formulae:                               u          =                                                                      S                  xy                                ⁢                                  S                  yz                                            -                                                S                  xz                                ⁢                                  S                  yy                                                                                                      S                  xx                                ⁢                                  S                  yy                                            -                              S                xy                2                                                    ⁢                  
                ⁢                  v          =                                                                      S                  xy                                ⁢                                  S                  xz                                            -                                                S                  yz                                ⁢                                  S                  xx                                                                                                      S                  xx                                ⁢                                  S                  yy                                            -                              S                xy                2                                                                        (2)            
with:                                                         Sxx              =                              xe2x80x83                            ⁢                                                ∑                                      i                    ∈                    F                                                  ⁢                                  Ex                  i                  2                                                                                                        Syy              =                              xe2x80x83                            ⁢                                                ∑                                      i                    ∈                    F                                                  ⁢                                  xe2x80x83                                ⁢                                  Ey                  i                  2                                                                                                        Sxz              =                              xe2x80x83                            ⁢                                                ∑                                      i                    ∈                    F                                                  ⁢                                  xe2x80x83                                ⁢                                                      Ex                    i                                    ⁢                                      Ez                    i                                                                                                                          Sxy              =                              xe2x80x83                            ⁢                                                ∑                                      i                    ∈                    F                                                  ⁢                                  xe2x80x83                                ⁢                                                      Ex                    i                                    ⁢                                      Ey                    i                                                                                                                          Syz              =                              xe2x80x83                            ⁢                                                ∑                                      i                    ∈                    F                                                  ⁢                                                      Ey                    i                                    ⁢                                      Ez                    i                                                                                                          (3)            
In the ideal case where the displacements of the pixels situated inside the window F are identical, that is to say when the constant-field model is entirely valid, the sum W is zero. In the perturbed cases of channel complexes, the displacements of the pixels situated inside the window F are different and then the sum W is non zero and it is in fact very big.
The purpose of the present invention is, on the one hand, to remedy the aforesaid drawbacks which reside in the fact that the calculation of the flux error is dependent on the direction of visualization and that calculating the error in all the directions of the seismic survey would be prohibitive, and on the other hand, to propose a method which makes it possible to detect chaotic structures in a reliable manner and with a considerably reduced calculation time.
The subject of the present invention is a method of detecting chaotic structures in a given medium, of the type consisting in:
a) representing the given medium by means of at least one sequence of images with axes x and y and arranged along a perpendicular axis z in such a way as to construct an image block with axes x, y and z,
b) defining a three-dimensional analysis window F with axes parallel to the axes x, y and z,
c) centering the window F on an image point of the block,
d) calculating the components along the x, y and z axes of the light intensity gradient vector E at every point of the window F,
characterized in that it furthermore consists in:
e) calculating an elementary matrix M at every point of the window F and representing the direct product Exc3x97ET where ET is the transpose of the gradient vector E,
f) summing the elementary matrices M for all the points of the window F, in such a way as to obtain a sum matrix A which is assigned to the said image point on which the window is centered,
g) diagonalizing the said sum matrix A so as to determine its eigenvalues xcex1, xcex2, xcex3, each eigenvalue corresponding to an eigenvector,
h) quantifying, at the image point, center of the window, the minimum global error in the optical flux vector U in an oriented direction with unit vector D, as a function of the said eigenvalues xcex1, xcex2 and xcex3 with the constraint UTxc3x97D=1 where UT is the transpose of the vector U,
i) eliminating the contribution of the largest eigenvalue in the quantification of the error in the optical flux in such a way as to obtain a quantification of a secondary error in the optical flux as a function of the two remaining eigenvalues for the said oriented direction;
j) defining a multidirectional error by integrating the secondary error in the plane defined by the eigenvectors corresponding to the remaining eigenvalues,
k) assigning the multidirectional error to the image point on which the window F is centred, and
l) calculating the multidirectional errors assigned to all the image points of the image block.
An advantage of the present invention resides in the fact that the calculated flux error is multidirectional, that is to say it is calculated in all the directions of a plane defined by the eigenvectors corresponding to the remaining eigenvalues.
According to another characteristic of the present invention, it is only necessary to determine the formula representative of the multidirectional error assigned to one image point and then to apply it in respect of the eigenvalues of each diagonalized matrix corresponding to each of the other image points of the block analysed.
In this way, the calculation time in respect of the entire set of image points is considerably reduced, thus making it possible to analyse ever larger and ever more complex seismic image blocks for example in times which are reasonable compared with those which are required for implementing the methods of the prior art.
According to another characteristic, the multidirectional errors are selected as a function of a threshold which can be determined or defined gradually, so that after having defined an appropriate threshold, the values of the multidirectional errors which are less than the said appropriate threshold are eliminated. Thus, the limits of the envelope determined by the points corresponding to a multidirectional error equal to the prescribed threshold, will correspond to the limits of the channel complex.